Comment on "Gaussian orthogonal ensemble statistics in a microwave stadium billiard with chaotic dynamics: Porter-Thomas distribution and algebraic decay of time correlations"

In a very interesting Letter, Alt et al. [1] have analyzed measurements on resonance widths of different eigen-modes of a superconducting cavity shaped like a stadium billiard, and have found agreement with a universal Porter-Thomas distribution (PT) [2]. However, one expects that the distribution in a stadium billiard is nonuniversal and should not obey Porter-Thomas due to the presence of bouncing ball (BB) orbits in this geometry. These orbits are well known to lead to deviations from random matrix theory (RMT) predictions for the eigenvalue spectra of chaotic geometries which have BB orbits, such as the stadium billiard [3] or the Sinai billiard [4]. Furthermore, they also lead to deviations from PT for the eigenfunction density distribution [5] in the Sinai billiard. Only geome-tries with no BB orbits, like the Sinai-Stadium, show direct experimental agreement with RMT, for the eigenvalue [4] and eigenfunction [5] statistics. In this Comment, we show that the agreement with PT claimed by Alt et al. [1] is because of the limited statistics and dynamic range of their experiment. Alt et al. measured the resonance widths of 950 eigen-modes of the stadium cavity, in a situation where the widths are dominated by the partial widths due to the coupling at three ports. Since this partial width is proportional to the wave-function amplitude, the experiment is a test of wave-function statistics for the density distribution P͑jcj 2 ͒. We have directly computed this distribution P͑jcj 2 ͒ by obtaining the numerical eigenfunction solutions to a stadium billiard of identical dimensions as in the experiment of Ref. [1], using a program provided by Heller [6]. Figure 1 shows two distributions corresponding to (a) the wave functions at three coupling positions as in the experiment for the 950 wave functions, and hence is a numerical simulation of the experiment of Ref. [1]; and (b) jcj 2 values on a lattice which is greater than the wavelength. In case (a) the numerical data appear to follow PT, although small deviations are present. This apparent agreement is because of the limited data set (2814 points) due to which there are only 21 points (out of the 2814 points) in the high intensity range jcj 2 ෇ 6 to 11 where deviations become more pronounced. More statistics can be generated by sampling the data at many more than three locations as in case (b), in which we sample about 1.5 3 10 …